For concreteness, consider the Glauber dynamics on the Hard-core model for \(G\)  whose adjacency matrix
has spectral radius \({\rho }\)  which is a bounded number. Let \({\lambda _c}(k)\)  by the critical value for the
Gibbs uniqueness of the Hard-core model on the infinite \(k\) -ary tree. We prove that the Glauber dynamics
mixes in \(O(n\log n)\)  steps for any fugacity \(0\le \lambda <{\lambda _c}({\rho })\) .
For a comparison, recall that the max-degree-\({\Delta }\)  bound for the Hard-core model comes from
[1]}, [2]} and requires fugacity \(0\le \lambda <{\lambda _c}({\Delta }-1)\)  in order
to get \(O(n\log n)\)  mixing time. This implies that our approach gives better bounds when the spectral
radius \({\rho }\)  is smaller than \({\Delta }-1\) . As a reference, note that we alway have that
\({\rho }\le {\Delta }\) . On the other hand, the spectral radius can get much smaller than the
maximum degree, e.g., for a planar graph we have that \({\rho }\le \sqrt{8{\Delta }-16}+2\sqrt{3}\) ,
see further cases in Section REF .
We get results of similar flavour for the Ising model, too.
